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Standard XIII

Mathematics

Chord of Contact

Points A and ...

Question

Points $A$ and $B$ lie on the parabola $y = 2 x^{2} + 4 x - 2$ , such that origin is the mid-point of the segment $A B$ . If $l$ is the length of the line segment $A B$ , then the value of $l^{2}$ is

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Solution

Given equation of the parabola
$y = 2 x^{2} + 4 x - 2$
Equation of the chord whose mid point is $(0, 0)$ will be
$T = S_{1} \Rightarrow 2 x \cdot 0 + 2 (x + 0) - 2 - \frac{(y + 0)}{2} = 2 (0)^{2} + 4 (0) - 2 - 0 \Rightarrow y = 4 x \dots (1)$
Now solving $(1)$ with the parabola equation
$4 x = 2 x^{2} + 4 x - 2 \Rightarrow x = \pm 1 ∴ y = \pm 4$
Hence $l = \sqrt{(4 + 4)^{2} + (1 + 1)^{2}}$
$∴ l^{2} = 68$

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Chord of Contact

Standard XIII Mathematics

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Points A and B lie on the parabola y=2x2+4x−2, such that origin is the mid-point of the segment AB. If l is the length of the line segment AB, then the value of l2 is

Given equation of the parabola y=2x2+4x−2 Equation of the chord whose mid point is (0,0) will be T=S1⇒2x⋅0+2(x+0)−2−(y+0)2=2(0)2+4(0)−2−0⇒y=4x ⋯(1) Now solving (1) with the parabola equation 4x=2x2+4x−2⇒x=±1∴y=±4 Hence l=√(4+4)2+(1+1)2 ∴l2=68

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