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Tangent Circles

The normal ch...

Question

The normal chord $P Q$ of a parabola $y^{2} = 4 a x$ at the point $(p, p)$ subtends a right angle at the focus.
If coordinate of $Q$ is $(u a, v a)$ , then find the value of $u + v$

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Solution

The correct option is D $3$
Since $x = y$ for $(p, p)$
$∴ a t_{1}^{2} = 2 a t_{1} o r t_{1} = 2$
$∴$ P is $(4 a, 4 a) .$
Now for normal chord
$t_{2} = - t_{1} - \frac{2}{t_{1}} = - 3$
$∴$ Q is $(a t_{2}^{2}, 2 a t_{2}) = (9 a, - 6 a)$
$∴ u + v = 9 - 6 = 3$

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Similar questions

If normal at p to parabola $y^{2} = 4 a x$ meets parabola again at Q such that PQ subtends a right angle at the vertex of $y^{2} = 4 a x$ then p will be

Q. Let

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The normal chord PQ of a parabola y2=4ax at the point (p,p) subtends a right angle at the focus.If coordinate of Q is (ua,va), then find the value of u+v

The correct option is D 3Since x=y for (p,p)∴at21=2at1 or t1=2∴ P is (4a,4a). Now for normal chordt2=−t1−2t1=−3∴ Q is (at22,2at2)=(9a,−6a)∴u+v=9−6=3

The normal chord $P Q$ of a parabola $y^{2} = 4 a x$ at the point $(p, p)$ subtends a right angle at the focus.
If coordinate of $Q$ is $(u a, v a)$ , then find the value of $u + v$

The correct option is D $3$
Since $x = y$ for $(p, p)$
$∴ a t_{1}^{2} = 2 a t_{1} o r t_{1} = 2$
$∴$ P is $(4 a, 4 a) .$
Now for normal chord
$t_{2} = - t_{1} - \frac{2}{t_{1}} = - 3$
$∴$ Q is $(a t_{2}^{2}, 2 a t_{2}) = (9 a, - 6 a)$
$∴ u + v = 9 - 6 = 3$