Chord of Contact

Q. { 124. Lines }(1+λ)x+(4-λ)y+(2+λ)=0 and }(4-λ)x-}{(1+λ)y+(6-3λ)=0 are concurrent at points }A and }B}{ respectively and intersect at }C then locus of centroid of }}{Δ ABC is }(λ is parameter)

Q.

If the line y = mx + c be a tangent to the circle $x^2+y^2=a^2$, then the point of contact is

1. 

2. 

3. 

4. 

Q.

Equation of the pair of tangents drawn from the origin to the circle $x^2+y^2+2gx+2fy+c=0$ is

1. 

2. 

3. 

4. 

Q. 20. Equation of the circle which is such that the lengths of the tangents to it from the points (1, 0), (0, 2)and (3, 2) are 1, 7 and 2 respectively is

Q. 8. if chord of contact of tangent drawn from point p, q to ellipse x2/a2+y2/b2 =1 touches circle x2+y2=k2, then find locus of point p, q

Q.

What is the locus of the midpoint of chords of ($x-1{)}^2+y^2=1$ that passes through the origin.

1. x² + y² - x = 0

2. x² + y² - 3x = 0

3. x² + y² - y = 0

4. x² + y² - 3y = 0

Q. A variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to line $y=x$. Then the locus of point of intersection of normals at $P$ and $Q$ is:

1. $2x-y-12a=0$
2. $2x+y-a=0$
3. $x-2y-4a=0$
4. $3x-y-8a=0$

Q.

P and Q are 2 external points from which tangents are drawn to circle centered at origin and radius 'r' $P\equiv (x_1,y_1),Q\equiv (x_2,y_2).$ What is the condition for the lengths of tangents to be the same?

1. 

2. 

3. 

4. 

Q.

Find the equation of the chord of contact of tangents to the parabola
$y^2=4x$ from the point $P(3,4)$.

1. $x-2y-6=0$

2. $x+2y+6=0$

3. $x+y+3=0$

4. $x-2y-3=0$

Q. Tangents $PA$ and $PB$ are drawn to $x^2+y^2=4$ from the point $P(3,0)$. Area of $△PAB$ is equal to

1. $\frac{5}{9}\sqrt{5}$ sq unit
2. $\frac{1}{3}\sqrt{5}$ sq unit
3. $\frac{10}{9}\sqrt{5}$ sq unit
4. $\frac{20}{3}\sqrt{5}$ sq unit

Q. $TP$ and $TQ$ are tangents to the parabola $y^2=4ax$ at $P$ and $Q.$ If the chord $PQ$ passes through the fixed point $(-a,b),$ then the locus of $T$ is

1. $ay=2b(x-b)$
2. $bx=2a(y-a)$
3. $by=2a(x-a)$
4. $ax=2b(y-b)$

Q. If the chord of contact of the tangents from a point $P$ to the parabola $y^2=4ax$ touches the parabola $x^2=4by$, then the locus of $P$ is

1. $xy=-2ab$
2. $xy=ab$
3. $xy=2ab$
4. $xy=-ab$

Q. If the circles x² + y² = a and x² + y² − 6x − 8y + 9 = 0, touch externally, then a =
(a) 1
(b) −1
(c) 21
(d) 16

Q. Tangents are drawn from the point $P(2,2)$ to the circle $x^2+y^2=1,$ touching the circle at $A$ and $B.$ Then equation of circumcircle of $△PAB$ is

1. $x^2+y^2+2x+2y=0$
2. $x^2+y^2-2x-2y=0$
3. $x^2+y^2+2x-2y=0$
4. $x^2+y^2-2x+2y=0$

Q. Angle between the tangents to curves y = sinx and y = cosx at x = $\frac{\pi }{4}$ is

x = $\frac{\pi }{4}$ पर वक्रों y = sinx तथा y = cosx की स्पर्श रेखाओं के मध्य कोण है

1. ${\tan }^{-1}\left(2\sqrt{2}\right)$
2. ${\tan }^{-1}\left(\sqrt{2}\right)$
3. ${\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
4. $\frac{\pi }{4}$

Q. The tangents to the curve $y=(x-2{)}^2-1$ at its points of intersection with the line $x-y=3$, intersect at the point :

1. $(\frac{5}{2},-1)$
2. $(-\frac{5}{2},-1)$
3. $(\frac{5}{2},1)$
4. $(-\frac{5}{2},1)$

Q. The angle between pair of tangents drawn from any point on the circle $x^2+y^2=a^2$ upon the circle $x^2+y^2=b^2$ is $\frac{\pi }{3}$. Then, the locus of mid point of chord of contact is

1. $x^2+y^2=\frac{5a^2-b^2}{4}$
2. $x^2+y^2=\frac{5b^2-a^2}{4}$
3. $x^2+y^2=\frac{3a^2-b^2}{4}$
4. $x^2+y^2=\frac{4a^2-b^2}{4}$

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R$, then the line $lx+my+1=0$ touches a fixed circle whose tangents $PA$ and $PB$ are drawn to the above fixed circle from the points $P$ on the line $x+y-1=0$. Then chord of contact $AB$ passes through the fixed point

1. $(\frac{1}{2},\frac{-5}{2})$
2. $(\frac{1}{3},\frac{4}{3})$
3. $(\frac{-1}{2},\frac{3}{2})$
4. $(\frac{-1}{2},\frac{4}{3})$

Q. A circle of radius $\sqrt{5}$ units has diameter along the angle bisector of the lines $x+y=2$ and $x-y=2$. If chord of contact from the origin makes an angle of ${45}^{\circ }$ with the positive direction of $x$-axis, then the equation of the circle is

1. $(x+2{)}^2+(y-2{)}^2=5$
2. $(x-2{)}^2+(y+2{)}^2=5$
3. $(x-2{)}^2+y^2=5$
4. $(x+2{)}^2+y^2=5$

Q.

2 tangents $P{T}_1$ and $P{T}_2$ are drawn as shown in the figure and PQ passes through the centre O of the circle. Which segment(s) is/are the chord of contact of P.

1. RQ

2. $T_1{T}_2$

3. $T_{1}Qand{T}_{2}Q$

4. None of these

Q. Let x(x - a) + y(y - 1) be a circle. If two chords from (a, 1) bisected by the x-axis are drawn to the circle then the condition required is

1. $a^2$ = 8
2. $a^2$ $<$ 8
3. $a^2$ $>$ 8
4. a=0

Q. Tangents are drawn to the circle x²+y²= 16 at the points where it is met by the circle x²+y²-5x+3y-2=0, the point of intersection of these tangetnts is ?

Q. Tangents are drawn from different points on the line $x-y+10=0$ to the parabola $y^2=4x$. If the chords of contact pass through a fixed point, then the coordinates of the fixed point is

1. $(2,5)$
2. $(5,2)$
3. $(10,2)$
4. $(2,10)$

Q. Chord of contact $\left(L\right)$ from point $P$ on the circle $x^2+y^2-2x-4y-20=0$ makes an angle of ${60}^{\circ }$ at a point on the circumference of circle. If slope of the chord of contact is $\frac{1}{2},$ then

1. $L:x-2y=\frac{-(5\sqrt{5}+6)}{2}$
2. $L:x-2y=\frac{5\sqrt{5}-6}{2}$
3. $P\equiv (1+2\sqrt{5},2-4\sqrt{5})$
4. $P\equiv (1-2\sqrt{5},2+4\sqrt{5})$

Q. Tangent are drawn from the points on the line $x-y-5=0$ to $x^2+4y^2=4$, then all the chords of contact pass through a fixed point, whose coordinate are

1. $(\frac{4}{5},-\frac{1}{5})$
2. $(\frac{4}{5},\frac{1}{5})$
3. $(-\frac{4}{5},\frac{1}{5})$
4. None of the these

Q. The equation of the chord of contact of the tangents drawn from the point (3, 4) to the parabola $y^2=2x$ is .

1. x-4y+3=0
2. x-4y-3=0
3. x+4y+3=0
4. x+4y-3=0