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Intersection of a Line and Finding Roots of a Parabola

the line L1 p...

Question

the line L1 passing through the point(1,1) and the line L2 passes through the point(-1,1) .If the differenceof the slope of the lines is 2.Find the locus of the point of intersection of the line L1 and L2

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Solution

$Let the point of intersection be (x, y) Slope of line L 1 = \frac{y - 1}{x - 1} Slope of line L 2 = \frac{y - 1}{x + 1} Difference of slope is 2 . So, |\frac{y - 1}{x - 1} - \frac{y - 1}{x + 1}| = 2 \frac{y - 1}{x - 1} - \frac{y - 1}{x + 1} = \pm 2 When \frac{y - 1}{x - 1} - \frac{y - 1}{x + 1} = 2 \frac{(y - 1) (x + 1) - (x - 1) (y - 1)}{(x + 1) (x - 1)} = 2 yx + y - x - 1 - xy + x + y - 1 = 2 x^{2} - 2 2 y - 2 = 2 x^{2} - 2 x^{2} - y = 0 And \frac{y - 1}{x - 1} - \frac{y - 1}{x + 1} = - 2 \frac{(y - 1) (x + 1) - (x - 1) (y - 1)}{(x + 1) (x - 1)} = - 2 yx + y - x - 1 - xy + x + y - 1 = - 2 x^{2} + 2 2 y - 2 = - 2 x^{2} + 2 2 x^{2} + 2 y - 4 = 0 x^{2} + y = 2 So, locus of point is x^{2} - y = 0 and x^{2} + y = 2 .$

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Similar questions

Q. The line 'l_1' passing through the point

(1, 1)

and the 'l_2' passes through the point

(- 1, 1)

. If the difference of the slope of lines is

2

. Find the locus of the point of intersection of the

l_{1}

l_{2}

.

L 1

L 2

are perpendicular that intersect at the point

(2, 3)

L 1

passes through the point

(0, 2)

L 2

must pass through the point

L_{1}

passes through the point

(1, 1)

(2, 0)

and another line

L_{2}

(\frac{1}{2}, 0)

and perpendicular to

L_{1} .

Then the area (in sq. units) of the triangle formed by the lines

L_{1},

L_{2}

y -

L_{1}

passes through the points

(1, 1)

(2, 0)

and another line

L_{2}

(\frac{1}{2}, 0)

and is perpendicular to

L_{1} .

If the area of the triangle formed by the line

L_{1}, L_{2}

y -

\frac{p}{q}

sq. units where

p, q

are co-prime, then value of

p + q

L_{1} : \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{1}

L_{2} : \frac{x}{1} = \frac{y}{- 1} = \frac{z - 5}{3}

The equation of the line perpendicular to

L_{1}

L_{2}

and passing through the point of intersection of

L_{1}

L_{2}

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