Let the tangent drawn to the parabola y2=24x at the point \[(\alpha, \beta)\] is perpendicular to the line 2x+2y=5. Then the normal to the hyperbola \[\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\] at the point \[(\alpha+4, \beta+4)\] does NOT pass through the point :
Solutions · 1
Official solution
Any tangent to y2=24x at (\[\alpha\], \[\beta\])
\[\beta y = 12(x + \alpha )\]
Slope \[ = {{12} \over \beta }\] and perpendicular to 2x+2y=5
\[ \Rightarrow {{12} \over \beta } = 1 \Rightarrow \beta = 12,\,\alpha = 6\]
Hence hyperbola is \[{{{x^2}} \over {36}} - {{{y^2}} \over {144}} = 1\] and normal is drawn at (10, 16)
Equation of normal \[{{36\,.\,x} \over {10}} + {{144\,.\,y} \over {16}} = 36 + 144\]
\[ \Rightarrow {x \over {50}} + {y \over {20}} = 1\]
This does not pass though (15, 13) out of given option.